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BSE and TDDFT at work
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Claudio Attaccalitehttp://abineel.grenoble.cnrs.fr/http://abineel.grenoble.cnrs.fr/
BSE and TDDFT at work
CECAM Yambo School 2013 (Lausanne)CECAM Yambo School 2013 (Lausanne)
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Optical Absorption: Microscopic View
Direct and indirect interactions between an e-h pair created by a photon
Summing up all such interaction processes we get:
L(r1 t1 ; r2 t 2 ;r3 t 3 ;r4 t 4)=L(1,2,3,4)
The equation for L is the Bethe Salpeter Equation. The poles are the neutral excitations.
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1951
Bethe Salpeter Equation Historical remarks…
1970
First solution of BSE with dynamical effects: Shindo approximationShindo approximation JPSJ 29, 278(1970)
1974
First applications in solids: W. Hanke and L.J. Sham PRL 33, 582(1974) G. Strinati, H.J. Mattausch and W. Hanke
PRL 45, 290 (1980)
1995
Plane-waves implementationG. Onida et al.
PRL 75, 818 (1995)
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Feynman's diagrams andBethe-Salpeter equation
L(1234)=L0(1234)+
L0 1256 [v 57 56 78−W 56 57 68 ]L7834
Intrinsic 4-point equation.It describes the (coupled) propagation oftwo particles, the electron and the hole !
Quasihole and quasielectron
L=L0+ L0[v−W ]L
W 1,2=W r1 , r2 t 1 , t2Retardation effects are
neglected
+ -=
L1,2,3,4 =L r1, r2, r3, r4 ; t −t 0=L1,2,3,4,
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Construction of the BSE 1/2
L0(r1,r2,r '1,r '2∣ω)=2∫ dω '2π
G0(r1,r '2 ;ω+ω ' )G0(r2,r '1 ;ω ' )
G0(r1,r2 ;ω)=∑k , i
ϕk ,i∗ (r1)ϕk , i(r2)
ℏω−ϵk , i+iηsign(ϵk , i−ϵF)
where
We expand L in the independent particle basis
L(r1, r2,r '1,r '2∣ω)= ∑k1, k2, k3,k4
ϕk1(r1)ϕk2
(r2)Lk1, k2, k3,k 4ϕk3(r '1)ϕk4
(r '2)
We start from L0
...integrating in the frequency we get...
Lk1, k2, k '1,k '2
0=δk1, k2
δk '1, k ' 2
2iℏ
f (ϵk2)−f (ϵk ' 2
)
ϵk ' 2−ϵk2
+iδ+ℏω
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Construction of the BSE 2/2
Time-dependent Hartree term:
Screened Exchange Coulomb term:
...and now let's solve the equation...
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H ij , klTD−harteee=2∫ϕi
e(r )ϕ j∗h(r )V (r−r ')ϕk
e (r ')ϕl∗h(r ')
H ij , klTD−SEX=∫ϕi
e (r )ϕ j∗h(r ' )W (r , r ')ϕk
e (r )ϕl∗h(r ')
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Bethe-Salpeter equation (4-points - space and time)
-
+
-
+
-
+
We work in transition space...
Should we invert the equation for L for each frequency???
H n1n2 ,n3n4 exc A
n3 n4=E An1n2
L1,2,3,4 =L r1, r2, r3, r4 ; t −t 0=L1,2,3,4,
The frequency term can be separated and an-effective Hamiltonian can be derived without any frequency dependency
L=L0+L0[v−W ]L
[L0−1+[v−W ] ]=L−1 We solve the inverse Bethe-Salpeter
eq., because it is easier
Original BSE
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How to transform the BSE in an eigenvalues problem
Using the definition of L0
That is diagonal in the e/h space
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We can solve the equation once for all frequency!!!!
...with some linear algebra....
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Absoprtion spectra and BSE
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BSE calculation in practice
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… Some results…
V. Garbuio et al., PRL 97, 137402 (2006)
Bruneval et al., PRL 97, 267601 (2006)
Tiago et al., PRB 70, 193204 (2004)
Strinati et al., Rivista del Nuovo Cimento 11, 1 (1988)
Bruno et al., PRL 98, 036807 (2007)
Albrecht et al., PRL 80, 4510 (1998)
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Excitons in nanoscale systems
Excitons in nanoscale systemsGregory D. Scholes, Garry RumblesNature Materials 5, 683 - 696 (2006)
Nanotubes/Nanowires
Colloidal quantum dots
Frenkel excitons in photosynthesis
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BSE for charge transfer excitons
donor-acceptor complexes:
benzene, naphthalene, and anthracene
derivatives with the tetracyanoethylene
acceptorX. Blase and C. Attaccalite Apl. Phys. Lett. 99, 171909 (2011)
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Exciton analysis
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Solving the equations in a smart way ...
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R
The BSE can be large...... too large
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Tamm-Dancoff approximation
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The dielectric constant doesn't require too much information
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Let's come back to the original formula
ϵ2(ω ,q )=4 πℑ [⟨P∣ 1ω−H EXC+iη
∣P ⟩]we can write the dielectric constant as
...and ask the help of mathematicians...
∣P ⟩=limq→0 e
iqr
q∣0 ⟩
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Lanczos-Haydock method
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Lanczos-Haydock algorithm
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Lanczos-Haydock performance
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What about TDDFT?
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TDDFT versus BSE
L(1234)=L0(1234)+
+L0(1256)[v (57)δ(56)δ(78)−W (56)δ(57)δ(68)]L(7834)
χ(12)=χ0(12)+χ0(13)[v (34)+f xc ]χ(42)
BSE
TDDFT
BSE is a 4-points equation => unavoidable
TDDFT is a 2-points equation => that can be rewritten as a 4-point equation
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TDDFT in G-space
χG ,G '(q ,ω)=χG ,G '0 (q ,ω)+χG ,G2
0 (q ,ω)(vG2(q)+ f G2, G3
xc (q))χG3 ,G(q ,ω)
ϵG ,G'−1 (q ,ω)=δG ,G '+vG(q)χG ,G '(q ,ω)
ϵM(q ,ω)=
1ϵG=0,G '=0−1 (q ,ω)
Simple static fxc case:
Microscopic dielectric constant:
Macroscopic dielectric constant:
Advantages: 2-points eq. Disadvantages: the eqs. Has to be solved for each frequency
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TDDFT in e/h space
Time-dependent exchange correlation function:-
H ij , klTD−hartree=2∫ϕi
e(r )ϕ j∗h(r )V (r−r ' )ϕk
e (r ')ϕl∗h(r ' )
Time-dependent Hartree term:
H ij , klTD−EXC=∫ϕi
e(r )ϕ j∗h(r ) f xc(r , r ')ϕk
e (r ' )ϕl∗h(r ' ) f xc(r , r ')=
∂V xc(r )
∂ρ(r ' )
H ij , klTD−SEX=∫ϕi
e (r )ϕ j∗h(r ' )W (r , r ')ϕk
e (r )ϕl∗h(r ' ) BSE
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Beyond the Tamm-Dancoff approx.
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Tamm-Dancoff breakdown 1
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Tamm-Dancoff breakdown 2
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Don't worry! Haydock method still works
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SUMMARY
● Optical spectra can be calculated by mean of Green's function theory
●TDDFT and BSE can efficiently be formulated in the e/h space
● By using Lanczos-Haydock approach we do not need to diagonalize the full matrix!
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References!!!Reviews:● Application of the Green’s functions method to the study of the optical properties of semiconductors Nuovo Cimento, vol 11, pg 1, (1988) G. Strinati
● Effects of the Electron–Hole Interaction on the Optical Properties of Materials: the Bethe–Salpeter EquationPhysica Scripta, vol 109, pg 141, (2004) G. Bussi
● Electronic excitations: density-functional versus many-body Green's-function approachesRMP, vol 74, pg 601, (2002 ) G. Onida, L. Reining, and A. Rubio
On the web:● http://yambo-code.org/lectures.php● http://freescience.info/manybody.php● http://freescience.info/tddft.php● http://freescience.info/spectroscopy.php
Books: